Keith J. Devlin

In 2000, the Clay Foundation announced a historic competition: whoever could solve any of seven extraordinarily difficult mathematical problems, and have the solution acknowledged as correct by the experts, would receive 1 million in prize money. There was some precedent for doing this: In 1900 the mathematician David Hilbert proposed twenty-three problems that set much of the agenda for mathematics in the twentieth century. The Millennium Problems--chosen by a committee of the leading mathematicians in the world--are likely to acquire similar stature, and their solution (or lack of it) is likely to play a strong role in determining the course of mathematics in the twenty-first century. Keith Devlin, renowned expositor of mathematics and one of the authors of the Clay Institute's official description of the problems, here provides the definitive account for the mathematically interested reader. [Review by David Roberts, on 02/7/2003] In May 2000, the Clay Mathematics Institute elevated seven long-standing open problems in mathematics to the status of "Millennium Prize Problems," endowing each with a million-dollar prize. The seven particular problems were chosen in part because of their difficulty, but even more so because of their central importance to modern mathematics. The problems and the corresponding general areas of mathematics are as follows. 1) The Riemann Hypothesis - Number Theory 2) Yang-Mills Existence and Mass Gap - Mathematical Physics 3) The P versus NP problem - Computer Science 4) Navier-Stokes Existence and Smoothness - Mathematical Physics 5) The Poincaré Conjecture - Topology 6) The Birch and Swinnerton-Dyer Conjecture - Number Theory 7) The Hodge Conjecture - Algebraic Geometry The Navier-Stokes equations were first written down in the early 1820's, Riemann made his hypothesis in an 1859 paper, and the Poincaré conjecture dates from 1904. The remaining problems arose in the period 1950-1971. In The Millennium Problems, Keith Devlin aims to communicate the essence of these seven problems to a broad readership. It is, of course, a very ambitious goal. The preface makes it clear what Devlin's ground rules are. First he assumes only "a good high school knowledge of mathematics." Second, he is writing "not for those who want to tackle one of the problems, but for readers — mathematician and non-mathematician alike — who are curious about the current state at the frontiers of humankind's oldest body of scientific knowledge." He is clear that the readership drives the level of the book, so that precise statements of the problems will not always be given. Rather the goal is "to provide the background to each problem, to describe how it arose, explain what makes it particularly difficult, and give... some sense of why mathematicians regard it as important." After the short preface, the book has an interesting Chapter 0, and then one chapter for each problem in the above order. These seven chapters are constructed similarly. Most have a long historical component, generally including biographical information about the person or persons after whom the conjecture is named. Each has substantial background mathematical information, with topics ranging from complex numbers in Chapter 1 and group theory in Chapter 2 to congruences in Chapter 6 and algebraic varieties in Chapter 7. Applications are emphasized when possible. A nice theme in Chapters 2 and 4 is that mathematicians are behind physicists and engineers and just trying to catch up. Each chapter concludes with a discussion of the millennium problem itself. Chapter 5 illustrates how Devlin ties the various units of a chapter into a coherent narrative. It begins with four pages about the life and work of Henri Poincaré. It moves on to introduce "rubber sheet geometry" in terms of how subway maps and refrigerator wiring diagrams are not geometrically faithful to the physical objects they represent, but nonetheless clearly capture all relevant information. This unit is important as it will make readers feel that topology is natural, rather than weird. Chapter 5 next introduces the concepts of vertices, edges, faces and finally Euler characteristic in terms of the Königsberg bridge problem. It introduces non-orientable surfaces and makes the introduction of an ambient four-space seem natural, since it is necessary for an embedding of the Klein bottle. It topologically classifies closed surfaces first crudely in terms of their orientability, and then completely in terms of networks drawn upon them and the Euler characteristic of these networks. It gives a very attractive example of two seemingly linked rings that in fact can be pulled apart. This example shows the reader that not everything is geometrically obvious, and thus underscores the utility of algebraic invariants that can rigorously confirm that two objects are topologically different. It discusses how the ordinary two-sphere is characterized among all closed surfaces by having the property that any loop on it can be shrunk continuously to a point. Finally, by way of this two-dimensional analogy, it discusses the actual three-dimensional Poincaré conjecture. The strain imposed by the challenge of communicating all seven millennium problems to a broad readership naturally shows at times. In the Navier-Stokes chapter, for example, the background mathematical information presented is calculus and specifically differentiation. Readers are instructed that "dy/dx" is to be read "dee-wye by dee-ex." Some seven pages later, the Navier-Stokes equations themselves are presented. They are four coupled non-linear partial differential equations in four independent variables. The exposition is gentle, but readers new to calculus will only understand at a superficial level. The strain is felt somewhat more in Chapter 6 and particularly so in Chapter 7. But these various strains are unavoidable, and I think in general Devlin has done a very good job giving general readers a feel for the seven millennium problems. The Millennium Problems concentrates on the past and present of the problems, but it's also natural to wonder about their future! Can we expect to see some prizes handed out within our lifetimes? Devlin raises this question at the end of the various chapters, but always in a noncommittal way. His mention of the "twenty-fifth century" in the preface may incline some readers to be pessimistic. My personal feeling is that there are good reasons for optimism. I'll take this opportunity to put down my guess that the torrid pace of mathematical progress in the 21st century will include the solution of at least two of the millennium problems before 2020 and at least five before the end of the century. When solutions to the millennium problems do come, it would be nice if the general public recognized them for the monumental achievements that they will be. Books such as Keith Devlin's The Millennium Problems will help a great deal.

"In The Math Gene, mathematician Keith Devlin offers a breathtakingly new theory of language development that describes how language evolved in two stages and how its main purpose was not communication. He goes on to show that the ability to think mathematically arose out of the same symbol-manipulating ability that was so crucial to the very first emergence of true language.". "The Math Gene explains how our innate pattern-making abilities allow us to perform mathematical reasoning. Revealing why some people loathe mathematics, others find it difficult and a select few excel at the subject, Keith Devlin suggests ways in which we can all improve our mathematical skills."--BOOK JACKET.

In The Language of Mathematics, Keith Devlin reveals the vital role mathematics plays in our eternal quest to understand who we are and the world we live in. More than just the study of numbers, mathematics provides us with the eyes to recognize and describe the hidden patterns of life-patterns that exist in the physical, biological, and social worlds without, and the realm of ideas and thoughts within. Taking the reader on a wondrous journey through the invisible universe that surrounds us - a universe made visible by mathematics - Devlin shows us what keeps a jumbo jet in the air, explains how we are able to view a football game on TV, and describes the mathematics that allow us to predict the weather, the behavior of the stock market, and the outcomes of elections.

Keith Devlin chronicles scientists' centuries-old quest to discover the laws of thought, from the astonishingly adept efforts of the ancient Greeks, to the invention of the first primitive "thinking machine" in the late nineteenth century, to radical findings that are challenging the very notion that the mind follows logical rules. Devlin introduces a host of new findings showing that many ways of thinking that are perfectly rational are at the same time entirely illogical, and that the exquisite verbal tango of human communication has little to do with logical processing. We must begin to appreciate, Devlin argues, that our minds are intimately intertwined with the world around us, and that our feelings and perceptions, even our social norms, play crucial roles in the marvelously complex dance of human cognition.

Examines the seventeenth-century correspondence between Blaise Pascal and Pierre de Fermat that described a mathematical method that became the foundation of probability, and discusses how it later developed into the concept of risk management in the twenty-first century.

The companion to the hit CBS crime series Numb3rs presents the fascinating way mathematics is used to fight real-life crimeUsing the popular CBS prime-time TV crime series Numb3rs as a springboard, Keith Devlin (known to millions of NPR listeners as "the Math Guy" on NPR's Weekend Edition with Scott Simon) and Gary Lorden (the principal math advisor to Numb3rs) explain real-life mathematical techniques used by the FBI and other law enforcement agencies to catch and convict criminals. From forensics to counterterrorism, the Riemann hypothesis to image enhancement, solving murders to beating casinos, Devlin and Lorden present compelling cases that illustrate how advanced mathematics can be used in state-of-the-art criminal investigations.

Keith Devlin reveals the astonishing range of creative and powerful ways in which scientists, artists, athletes, medical researchers, and many others are using mathematics to explore our world and to enhance our lives. On this tour you will explore deep-sea volcanoes with oceanographer Dawn Wright, go behind the scenes of blockbuster movies with special-effects designer Doug Trumbull, and probe the strange lives of viruses with microbiologist Sylvia Spengler. Listen to astronomer Robert Kirshner describe how he is charting the curve of space; discover how biologist Mike Labarbara visualizes the way a Tyrannosaurus rex carried its massive frame; and, along with brain researcher Brad Hatfield, peer into the mind of an Olympic markswoman at the moment she takes a shot. Glimpse a future of wearable computers and silicon "butlers" with computer scientist Pattie Maes, and watch a lilac come to life on screen with "computer botanist" Przemyslaw Prusinkiewicz.

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From NPR's Math Guy, the story of Leonardo of Pisa, the medieval mathematician who introduced Arabic numbers to the West and helped launch the modern era. In 1202, a young Italian published one of the most influential books of all time, introducing modern arithmetic to Western Europe. Leonardo of Pisa (better known today as Fibonacci) had learned the Hindu-Arabic number system when as a teenager he traveled with his father, a customs official for Pisa, to North Africa, then one of the principal mercantile centers of Europe. Devised in India in the seventh and eighth centuries and brought to North Africa by Muslim traders, the Hindu-Arabic system (featuring the numerals 0 through 9) offered a much simpler method of calculation than the then-popular finger reckoning and cumbersome Roman numerals. Though written in scholarly Latin, Fibonacci's book Liber Abbaci (The Book of Calculation) was the first to recognize the power of the 10 numerals, and to aim them at the world of commerce. It spawned generations of popular math texts in colloquial Italian and other languages that made it possible for ordinary people to buy and sell goods, convert currencies, and keep accurate records more readily than ever before—helping transform the West into the dominant force in science, technology, and large-scale international commerce. Liber Abbaci and Fibonacci's other books made him the greatest mathematician of the Middle Ages. Yet despite the ubiquity of his discoveries, Leonardo of Pisa has largely slipped from the pages of history. He is best known today for discovering the "Fibonacci sequence" of numbers that appears with great regularity in biological structures throughout nature, and is used by some to predict the rise and fall of financial markets. Keith Devlin re-creates the life and enduring legacy of an overlooked genius, and in the process makes clear how central numbers and mathematics are to our daily lives. - Publisher.

Stanford mathematician and NPR Math Guy Keith Devlin explains why, fun aside, video games are the ideal medium to teach middle-school mathematics. Teachers, education researchers, and professional game developers who want to produce video games for mathematics education will learn exactly what is involved in designing and producing successful math education video games that foster critical mathematical thinking skills necessary for success in a global economy. - Back cover.